From the related art, various systems are known which warn a driver of a motor vehicle when he is inattentive, especially in those cases where, due to the driver being tired, briefly nodding off or being distracted (in the following summarized as inattentiveness), the vehicle leaves, or is in imminent danger of leaving, the traffic lane. One example of such a system is shown in German Patent Application No. DE 195 07 956 where sensor elements determine the distance of the vehicle from a lane marking and the time for reaching the markings is estimated as a function of the distance and by taking additional variables into account. The driver is alerted when he is in danger of straying from the traffic lane. Problematic in such system is the difficulty of precisely ascertaining the attention state of the driver. While the change in the distance of a vehicle to a lane marking does supply indications of the driver's inattentiveness, it is difficult to reliably distinguish between a “concentrating driver” and an “inattentive driver.” A large gray area exists in which neither inattention nor attentiveness of the driver may be assumed. As a consequence, the following problems result with respect to alerting the driver. If the driver is warned too early, he will encounter numerous false alarms, which are felt to be disruptive. If the driver is warned only when inattentiveness must be assumed with the highest degree of certainty, the danger exists that not all cases are covered in which the driver is inattentive.